Questions: A health inspector is reviewing the average daily patient wait times (in minutes) at a local clinic. The clinic claims that the mean wait time is 14 minutes. The health inspector suspects that this may not be accurate. The population of wait times is normally distributed. The inspector collects a random sample of daily wait times, with the following observations (in minutes): 17, 18, 11, 16, 14, 19, 9. Conduct a hypothesis test at the 10% level of significance to determine if there is enough evidence to reject the clinic's claim. Which of the following statements regarding the p value is correct at 5% significance level. 0.025<p-value <0.05 0.01<p-value <0.025 p -value <0.01 p-value >0.1

A health inspector is reviewing the average daily patient wait times (in minutes) at a local clinic. The clinic claims that the mean wait time is 14 minutes. The health inspector suspects that this may not be accurate. The population of wait times is normally distributed. The inspector collects a random sample of daily wait times, with the following observations (in minutes): 17, 18, 11, 16, 14, 19, 9. Conduct a hypothesis test at the 10% level of significance to determine if there is enough evidence to reject the clinic's claim.

Which of the following statements regarding the p value is correct at 5% significance level.

0.025<p-value <0.05
0.01<p-value <0.025
p -value <0.01
p-value >0.1

Transcript text: A health inspector is reviewing the average daily patient wait times (in minutes) at a local clinic. The clinic claims that the mean wait time is 14 minutes. The health inspector suspects that this may not be accurate. The population of wait times is normally distributed. The inspector collects a random sample of daily wait times, with the following observations (in minutes): 17, 18, 11, 16, 14, 19, 9. Conduct a hypothesis test at the $10 \%$ level of significance to determine if there is enough evidence to reject the clinic's claim. Which of the following statements regarding the p value is correct at $5 \%$ significance level. $0.025<\mathrm{p}$-value $<0.05$ $0.01<\mathrm{p}$-value $<0.025$ p -value $<0.01$ $p$-value $>0.1$

Solution

Solution Steps

Step 1: Calculate Sample Statistics

The sample data collected consists of the following wait times (in minutes): $ 17, 18, 11, 16, 14, 19, 9 $.

The sample size $ n $ is calculated as: \[ n = 7 \]

The sample mean $ \bar{x} $ is calculated as: \[ \bar{x} = \frac{17 + 18 + 11 + 16 + 14 + 19 + 9}{7} = 14.8571 \]

The sample standard deviation $ s $ is calculated as: \[ s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} = 3.7161 \]

Step 2: Calculate Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} = \frac{3.7161}{\sqrt{7}} = 1.4046 \]

Step 3: Calculate Test Statistic

The test statistic $ t $ for the hypothesis test is calculated as: \[ t = \frac{\bar{x} - \mu_0}{SE} = \frac{14.8571 - 14}{1.4046} = 0.6103 \]

Step 4: Calculate P-value

For a two-tailed test, the p-value $ P $ is calculated as: \[ P = 2 \times (1 - T(|t|)) = 0.5641 \]

Step 5: Conclusion

At the $ 10\% $ significance level, we compare the p-value with the significance level. Since: \[ P = 0.5641 > 0.1 \] we do not reject the null hypothesis. This indicates that there is not enough evidence to reject the clinic's claim that the mean wait time is $ 14 $ minutes.